Let S be a set of n elements, then find out the number of ordered pairs in the largest and the smallest equivalence relations on S

The problem is asking about equivalence relations on a set \( S \) with \( n \) elements, specifically for the largest and the smallest equivalence relations. Let's break this down:

### Equivalence Relation
An equivalence relation on a set is a relation that is:
1. **Reflexive** (every element is related to itself),
2. **Symmetric** (if \( a \) is related to \( b \), then \( b \) is related to \( a \)),
3. **Transitive** (if \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \)).

### Number of Ordered Pairs in a Relation
In general, a relation on a set \( S \) of \( n \) elements can be thought of as a subset of the Cartesian product \( S \times S \), which consists of all ordered pairs \( (a, b) \), where \( a, b \in S \). The total number of ordered pairs in any relation is the size of the relation, i.e., the number of elements in the relation.

### Smallest Equivalence Relation
The smallest equivalence relation is the **identity relation**, where each element is only related to itself. This is because:
- It is reflexive (each element is related to itself),
- It is trivially symmetric and transitive (since there are no other pairs besides \( (a, a) \)).

For a set \( S \) with \( n \) elements, the number of ordered pairs in the smallest equivalence relation is simply the set of pairs where each element is related to itself. Hence, the number of ordered pairs in the smallest equivalence relation is \( n \) (one pair for each element: \( (a_1, a_1), (a_2, a_2), \dots, (a_n, a_n) \)).

### Largest Equivalence Relation
The largest equivalence relation is the **universal relation** where every element is related to every other element. In this case, the relation consists of all possible pairs of elements in \( S \). Since the relation must be reflexive, symmetric, and transitive, this means:
- Every element is related to every other element in the set.

Thus, the number of ordered pairs in the largest equivalence relation is equal to the total number of pairs in \( S \times S \), which is \( n^2 \) (since there are \( n \) elements, each element can be paired with \( n \) elements, including itself).

### Conclusion
- The number of ordered pairs in the **smallest equivalence relation** is \( n \).
- The number of ordered pairs in the **largest equivalence relation** is \( n^2 \).

Would you like more details on this explanation?

Here are 5 related questions:
1. What are the conditions for a relation to be transitive?
2. How do you determine if a relation is symmetric?
3. Can there be more than one equivalence relation on a set with \( n \) elements?
4. How many equivalence classes are there in the largest equivalence relation?
5. What is the number of ordered pairs in a partial equivalence relation?

**Tip:** An equivalence relation partitions a set into disjoint equivalence classes where every element in a class is related to every other element in that class.

Let S be a set of n elements, then find out the number of ordered pairs in the largest and the smallest equivalence relations on S.

Explore how to calculate the number of ordered pairs in both the largest and smallest equivalence relations for a set S with n elements. This problem includes the concept of equivalence relations, set theory, and the Cartesian product, with step-by-step explanations of the identity and universal relations. Ideal for undergraduate students studying set theory.

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